Independent resolutions for totally disconnected dynamical systems. I: Algebraic case (Q2512709)

Given an arbitrary dynamical system on a totally disconnected space, the authors attach to it a sequence of totally disconnected dynamical systems which all admit invariant regular bases and whose \(\mathbb{Z}\)-algebras (the algebra of continuous integer-valued functions) give rise to a resolution of the \(\mathbb{Z}\)-algebra of the original dynamical system (or rather of its underlying space). The main feature is that everything works equivariantly with respect to the group actions. The authors call such a resolution an algebraic independent resolution. This notion leads to \(C^*\)-algebraic independent resolutions for the purpose of \(K\)-theoretic computations for the crossed product of the original dynamical system. The paper contains also concrete examples showing how to compute group homology and cohomology using algebraic independent resolutions.